CORRIGENDUM TO “(ALMOST) EVERYTHING YOU ALWAYS WANTED TO KNOW ABOUT DETERMINISTIC CONTROL PROBLEMS IN STRATIFIED DOMAINS”

. The aim of this short note is: ( i ) to report an error in [1]; ( ii ) to explain why the comparison result of [1] lacks an hypothesis in the deﬁnition of subsolutions if we allow them to be discontinuous; ( iii ) to describe a simple counter-example; ( iv ) to show a simple way to correct this mistake, considering the classical Ishii’s deﬁnition of viscosity solutions; ( v ) ﬁnally, to prove that this modiﬁcation actually ﬁxes the the comparison and stability results of [1].


1.
Introduction. The aim of this note is to report and correct an error that we have found in [1]. We illustrate the problem we are facing by producing an explicit counter-example to the comparison result but we also solve this difficulty by updating the definition of subsolutions (no modification is needed for the supersolution condition). To give the reader a quick (yet precise) formulation of the corrections we develop hereafter, let us summarize them as 1. All the results in [1] are valid, as they are formulated, if the subsolutions are assumed to be continuous functions in R N . 2. All the results in [1] are also valid in the case of upper semi-continuous subsolutions, provided we assume that they satisfy an Ishii's subsolution condition in addition to the "stratified" definition found in [1]. Let us now give more details. In [1], we are considering deterministic control problems whose dynamics and costs (b, l) at any point (x, t) ∈ R N × [0, T ] are chosen in a bounded, closed and convex set BL(x, t). The classical Hamiltonian is defined by H(x, t, p) := sup and the aim is to give a suitable sense and study the associated Hamilton-Jacobi-Bellman Equation which, for classical problems, reads

GUY BARLES AND EMMANUEL CHASSEIGNE
If, for such classical problems, BL(x, t) and H are continuous everywhere, having a stratified problem means on the contrary that they can have discontinuities but with a particular structure. More precisely, R N can be decomposed as where, for any k, M k is a k-dimensional submanifold of R N . The manifolds M 0 , M 1 , . . . , M N −1 are disjoints (but they may have several connected components) and they are the locations where either BL(x, t) or H can have discontinuities, which essentially means (taking into account the assumptions made on BL) that we have more dynamics and costs on each M k , which can be seen as specific control problems on M k . To treat in a proper way these specific control problems on M k , it is necessary to introduce the Hamiltonians which are defined on M k by where T x M k is the tangent space to M k at x and the associated Hamilton-Jacobi-Bellman Equation The first aim of [1] was to provide a definition of viscosity sub and supersolution for Hamilton-Jacobi-Bellman Equations in Stratified Domain [(HJB-SD) for short], namely (1)- (2).
Before going further, we point out that we assume throughout this short note that the natural assumptions for a stratified problem are always satisfied: (i) M = (M k ) k=0..N is a regular stratification of R N , (ii) the Hamiltonians (or BL) satisfy the key assumptions of [1], namely (TC) (tangential continuity), (NC) (normal controllability) and (LP) (Lipschitz continuity).
The definition of super and subsolutions in [1] follows the ones of Bressan & Hong [2]: a lower semi-continuous function v is a supersolution of (HJB-SD) if it is a supersolution of (1) in the classical Ishii's sense, while an upper semi-continuous function u is a subsolution of (HJB-SD) if it is a subsolution (again in the classical Ishii's sense) of each equation (2) for any k = 0, · · · , N . And it is worth pointing out that these H k -inequalities are really inequalities on M k × (0, T ], i.e. they are obtained by considering maxima of u − φ on M k × (0, T ] for any smooth testfunction φ.
Unfortunately, this way of defining subsolutions only in terms of the H k 's is not sufficent since it treats all the M k separately without linking them and this allows u to have artificial values on certain manifolds M k . Let us mention that this difficulty does not appear in Bressan & Hong [2] since the subsolutions are assumed to be continuous.
A counter-example -Consider in R N the equation for which the "control" solution is given, in B(0, 1) by where D T u stands for the tangent derivative of u on the sphere S(0, 1). For this stratified formulation, As we said above, the key fact in this counter-example to comparison is that we can put artificial values on M N −1 for the subsolution u since these values are unrelated with those of u on M N . If u is assumed to be continuous, then the only solution is U .
We also point out that we could have provided a more pathological counter- and for any k = 0..N , We recall that 3. Correcting the approximation argument. A key step in the comparison proof in [1] is to regularize u by a tangent sup-convolution and then by a standard convolution. We claim in [1, Lemma 5.5] that, by doing so, we obtain a sequence of subsolutions (u ε ) ε which are continuous but the above counter-example shows that this statement was wrong in general. The reason is that if x ∈ M k and u(x) > lim sup u(y) : y → x y / ∈ M k , then the tangential sup-convolution need not be continuous in the normal direction to M k . The fix consists in making sure that the following property is fulfilled (a) For any where, for r > 0 small enough, U + , U − ⊂ M N ∩ B(x, r) are the locally disjoint connected components of R N \ M N −1 ∩ B(x, r). Lemma 5.5 in [1] can now be corrected by assuming (a) and (b) Lemma 3.1. Let x ∈ M k and t, h > 0. There exists r > 0 such that if u is a subsolution of (HJB-SD) in B(x, r ) × (t − h, t) satisfying in addition conditions (a) and (b), then for any a ∈ (0, r ), there exists a sequence of Lipschitz continuous Proof. The proof is exactly the one given in [1] but we need a little additional argument. We first reduce to the case of a flat stratification through a change of variables. Without loss of generality, we can assume that x = 0, and writing the coordinates in R N as (y 1 , y 2 ) with y 1 ∈ R k , y 2 ∈ R N −k we may assume that M k := {(y 1 , y 2 ) : y 2 = 0}.
Then we perform a sup-convolution in the M k directions (and also the time direction) by setting for some large enough constant K > 0 (as explained in [1], for k = 0 it is enough to do only a time sup-convolution).
Here is the place where we introduce an additional regularity argument: the sup-convolution is clearly is Lipschitz continuous in the y 1 and s variables and the normal controllability implies that it is also Lipschitz continuous in the y 2 -variable for y 2 = 0. It remains to connect the values of u ε1,α1 1 (y 1 , y 2 , s) for y 2 = 0 and y 2 = 0; this is where we need conditions (a) and (b). With this addition, the rest of the proof remains identical.
Let us notice that in particular, Lemma 3.1 is valid if u is continuous since (a) and (b) are obviously satisfied. This is why there is no problem at all in [1] if we consider continuous subsolutions. Now, in the case of upper semi-continuous subsolutions, a more usable assumption than (a) − (b) is to consider viscosity subsolutions in the sense of Ishii. Proof. In order to prove (a) we can assume that we are in the stationary case (for simplicity) and that M k if flat (this reduction is done in [1] using the regularity of M k ). Consider the function where e is any unit vector normal to M k . If u(x) > lim sup{u(y), y → x, y / ∈ M k }, the maximum of this function is necessarily achieved on M k at y ε . Using the normal controllabillity in M N we deduce that H * is coercive in any normal direction to M k so that if C is large enough we reach a contradiction in H * (y ε , u(y ε ), Dφ(y ε )) ≤ 0, which gives the desired property (i).
In the case of M N −1 , the same proof works but this is not enough since, locally, R N \ M N −1 has two connected components, U + and U − and we have to show that the property is true separately for both connected components. Here, we assume again that M N −1 is an hyperplane and we take the same test-function but with e = +n or −n where n is a normal vector to M N −1 .
If u(x) > lim sup{u(y), y → x, y / ∈ M N −1 , n · (y − x) > 0}, we consider The maximum cannot be achieved in the domain where n · (y − x) > 0 because of the above hypothesis on u(x). But in the complementary of this set, the term +Cn · (y − x) has the right sign (i.e., it is non-positive), allowing to show that a maximum is achieved and is converging to x as ε → 0. Therefore we can choose C ε −1 and again the normal controllability of H * allows to get the contradiction.  For any open subset Ω of R N and for any 0 ≤ t 1 < t 2 ≤ T , we have a comparison result for (HJB-SD) in Q = Ω × (t 1 , t 2 ), i.e. for any bounded upper semi-continuous stratified subsolution u of (HJB-SD) in Q and any bounded lower semi-continuous supersolution v of (HJB-SD) in Q, then where ∂ p Q denotes the parabolic boundary of Q, i.e. ∂ p Q : The immediate Corollary is that there is a unique stratified solution of the problem. Concerning the stability result [1, Theorem 6.2], we need only to modify the subsolution part as follows Theorem 4.2. Assume that (HJB − SD) ε is a sequence of stratified problems associated to sequences of regular stratifications (M ε ) ε and of Hamiltonians (H ε , H k ε ) ε . If M ε RS −→ M, then the following holds (i) if for all ε > 0, v ε is a lower semi-continuous supersolution of (HJB − SD) ε , then v = lim inf * v ε is a lower semi-continuous supersolution of (HJB-SD), the HJB problem associated with H = lim sup * H ε . (ii) If for ε > 0, u ε is a stratified upper semi-continuous subsolution of (HJB − SD) ε and if the Hamiltonians (H k ε ) k=0..N satisfy (NC) and (TC) with uniform constants, thenū = lim sup * u ε is a stratified upper semi-continuous subsolution of (HJB-SD) with H k = lim inf * H k ε for any k = 0..N . Proof. The proof is identical to the one in [1], the only new argument we need to add concerns the fact that the Ishii condition is stable as ε → 0: if u ε is the subsolution satisfying (H ε ) * (x, u ε , Du ε ) ≤ 0 in R N , then u := lim sup * u ε also satisfies the limit Ishii condition H * (x, u, Du) ≤ 0 in R N .
Then, Corollary 1 in [1] immediately follows in the class of stratified solutions.